Animal cells and plant cells have very different mechanisms of mitosis. This is mainly due to the structural makeup of the different types of cells. For example, the cell wall affects how plant cells undergo mitosis.
Within animal cells, organelles called centrosomes assist organelles called asters and spindle fibers in pulling apart the sister chromatids to opposite poles of the sister cells. In plant cells, these asters are not present, and the spindle fibers pull apart the chromatids on their own.
The process of cytokinesis, or the splitting of the cells, is also vastly different between the two types of cells. In animal cells, a cleavage furrow forms between the two forming cell bodies after DNA and organelles have been replicated and situated in their new cell bodies. This furrow continues to close until the cells separate and become two separate bodies. In plant cells, in place of a cleavage furrow, a cell wall begins to form down the center of the cell, separating the polar bodies. Once the cell wall is finished forming, the cell has fully separated into two, even though it shares a cell wall.
These concepts can grow more advanced and elaborate depending on the cell type, for example, single celled organisms versus human cells.